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单词 ExpectedValue
释义

expected value


Let us first consider a discrete random variable X with values in . Then X has values in an at most countable set 𝒳. For x𝒳 denote the probability that X=x by Px. If

x𝒳|x|Px

converges, the sum

x𝒳xPx

is well-defined. Its value is called the expected valueMathworldPlanetmath, expectation or mean of X. It is usually denoted by E(X).

Taking this idea further, we can easily generalize to a continuous random variable X with probability density ϱ by setting

E(X)=-xϱ(x)𝑑x,

if this integral exists.

From the above definition it is clear that the expectation is a linear function, i.e. for two random variables X,Y we have

E(aX+bY)=aE(X)+bE(Y)

for a,b.

Note that the expectation does not always exist (if the corresponding sum or integral does not converge, the expectation does not exist. One example of this situation is the Cauchy random variable).

Using the measure theoretical formulation of stochastics, we can give a more formal definition. Let (Ω,𝒜,P) be a probability space and X:Ω a random variable. We now define

E(X)=ΩX𝑑P,

where the integral is understood as the Lebesgue-integral with respect to the measure P.

Titleexpected value
Canonical nameExpectedValue
Date of creation2013-03-22 11:53:42
Last modified on2013-03-22 11:53:42
Ownermathwizard (128)
Last modified bymathwizard (128)
Numerical id21
Authormathwizard (128)
Entry typeDefinition
Classificationmsc 60-00
Classificationmsc 46L05
Classificationmsc 82-00
Classificationmsc 83-00
Classificationmsc 81-00
Synonymmean
Synonymexpectation value
Synonymexpectation
Related topicAverageValueOfFunction
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